Menu for bevel edges
I don't think it's possible at the moment: if you start bevelling using Ctrl+B you are forced to use the keyboard-based workflow.

However, you can use the bevel tool (green icon in the left-side toolbar) instead of the bevel operation, which gives you access to some more options (not all the shortcuts though).

In combination with the "tool settings" header, it allows you to change the options right before starting your bevel operation:





Actions performed in the above gif:


Edit mode, select your edge
Select the Bevel tool
If it's not active already, activate the "Tool Settings" header by right-clicking on the 3D Viewport header
Adjust the setting
Hold down the mouse button and drag




Alternatively to the "Tool settings" toolbar, you can display a "Tool properties" panel in the Properties area.